The Embedding Flow of 3-Dimensional Locally Hyperbolic \(C^\infty \) Diffeomorphisms
- 149 Downloads
For finite dimensional locally hyperbolic \(C^\infty \) smooth real diffeomorphisms, the problem on the existence and smoothness of their embedding flows was solved by Li et al. (Am J Math 124:107–127, 2002) in the weakly non-resonant case. Whereas there are no general results on the existence of embedding flows in weakly resonant cases, except for some examples and some ones on the non-existence of embedding flows. The simplest hyperbolic diffeomorphisms having weakly resonant phenomena should be the 3-dimensional ones. This paper will concentrate on the embedding flow problem for this kind of diffeomorphisms. In one case we obtain complete characterization of the diffeomorphisms which admit embedding flows, and in the other cases we obtain some necessary conditions for the diffeomorphisms to have embedding flows. Also in this latter case we provide a sufficient condition for the existence of \(C^\infty \) embedding flow. As we know it is the first general result on the existence of embedding flows for locally hyperbolic \(C^\infty \) diffeomorphisms with weak resonance. These results show that the embedding flow problem in the weakly resonant case is much more involved.
Keywords3-Dimensional local diffeomorphism Weak resonance Hyperbolicity \(C^\infty \) smoothness Embedding flow
Mathematics Subject Classification34A34 34C20 34C41 37G05
The author is partially supported by NNSF of China grants 10831003 and 11271252, RFDP of Higher Education of China grant 20110073110054 and FP7-PEOPLE-2012-IRSES-316338 of Europe, and by innovation program of Shanghai municipal education commission grant 15ZZ012.
- 4.Beyer, W.A., Channell, P.J.: A Functional Equation for the Embedding of a Homeomorphim of the Interval Into a Flow. Lecture Notes in Math, vol. 1163. Springer, New York (1985)Google Scholar
- 10.García, I.A., Giacomini, H., Grau, M.: The inverse integrating factor and the Poincaré map. Trans. Am. Math. Soc. 362, 1612–3591 (2010)Google Scholar
- 12.Kuksin, S., Pöschel, J.: On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications. In: Kuksin, S., Lazutkin, V., Pöschel, J. (eds.) Seminar on Dynamical Systems, pp. 96–116. Birkhäuser, Basel (1978)Google Scholar
- 15.Li, Weigu: Normal Form Theory and its Applications (in Chinese). Science Press, Beijing (2000)Google Scholar
- 20.Voronin, S. M.: The Darboux–Whitney theorem and related questions, In Nonlinear Stokes phynomena of Adv. Soviet Math. 14, 139–233. American Mathematical Society, Providence (1993)Google Scholar